Calculating Bridge's Span

• 1). Go to the side of a bridge's support.
  
  
• 2). Measure the width of the support and divide that number by two. This is the center point of the support. Attach a string to this point of the support.
  
  
• 3). Bringing the free end of your string, go to the bridge's next support and find its center point, following Steps 1 and 2 above. Attach or mark the string at this point.
  
  
• 4). Measure the length of your string from the first end to the mark. This is the bridge's span.
  
  

Calculating the Arch

• 1). Find the points on either side of the arch where the curve ends. If the arch in question ends at, or intersects, with the actual crossway of the bridge, measure from one base of the arch to the next. If you are calculating for the arches that support the bridge from beneath in between the supports, use a level to find where the support ends and where the curvature of the arch begins on each side. Measure the distance between these two points. Write down the number as your "chord" length, c.
  
  
• 2). Divide that number by two, and use this to mark the halfway point of the arch. If you're dealing with the arches of the supports below, you can mark this on a string or length of tape hung from one end of the arch.
  
  
• 3). Record the distance from the chord's half way point to the top of the arch for your arc height, h. If you're dealing with an arch that is too big to measure on your own, there will likely be records of these measurements through the local government.
  
  
• 4). Using the two variables you wrote down for the chord length and the arch height, plug them into the following equation to get a radius (represented as "r") value. This is the radius of the invisible circle that would exist were the curvature of the arch extended all the way.
  
  r=(h^2 + (c^2)/4)/2h
  
  
• 5). Now that you have the radius you will need to calculate the angle (represented as "θ") at the center of the invisible circle that the arch would be a part of. If you imagine an upside down isosceles triangle with the chord and the radii of the circle extending down from the chord's ends to the center point of the circle, the angle at the bottom of the triangle and opposite the arch is the one you are going to calculate. To do this plug your values for the radius (r) and the chord (c) into the following equation using a scientific calculator:
  
  θ=arccosine(1-c^2/(2r^2))
  
  Make sure your calculator is giving answers in radians. (If your answer is over 7.3 then it is not.)
  
  
• 6). Multiply your answer by the value of the radius and this will be the value of your arc length.